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Proof step 2 from proof no-dups-update-key (previous step)

kar[m1] = key-index(key0, m0), no-dups-idx(kar, m) ⊦ no-dups-idx(kar[m1, key-index(key, m0)], m)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
kar[m1, key-index(key, m0)][m2].idx = kar[m1, key-index(key, m0)][m3].idx, kar[m1] = key-index(key0, m0), m3 < m, m3 < # kar, m2 < m, m2 < # kar, ∀ m0, m1. m0 < # kar ∧ m0 < m ∧ m1 < # kar ∧ m1 < m ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx ⊦ m2 = m3

Rule argument

Additional information

Used simplifier rules: no-dups-idx: ⊦ no-dups-idx(kar, n) ↔ (∀ m0, m1. m0 < # kar ∧ m0 < n ∧ m1 < # kar ∧ m1 < n ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx);% ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ # kar[n, ki] = # kar; (s)